Finol Exom

1'{A}The ptab- ihal an'Americon industry will lccote in ttaypllli: o.7 , the prcb. tlrot it vtil! lctole iti Brusseis is o.i ,' ond prob- thot it wiltlocote)p-eitlqr Munich c,, arffiiil-l o.8.whoi is rhe prob. thct t!.re incustry will locote inL'both cities ?. z-Mtyn$2 3.neiiher cities ? l.Munich siy=::!:!it has npt locate,c in Brussets ?t'l/E' ;(-r-,)t)f A,B,ondic areindepencentevents,show thoi:"Thetwoeveiits A'and BUC ieiisoindependent ,,.-.

\:=:-':'/:

z-{Afind the meoi p and lhe standortl deviation q oy d" nurnber oj ie!-ec.tiu.es X in o somple of 6ao pans' mr:ce by o michine , when the prolt- is o.o4 tr.Jt afiy one o; tnu prffirHtffi" .Then find on estinate foi .each ane ol tiie follawing tvvo probobitities: (7ibilx- y-l-_g end (z) pi 12 . y .:38 ) .

(B)The nuniber ol spotted imperfections in on, ,#rr", ,,::5,1, , poissqn nanndom voricbre wlth cn ,r#r*

_ 5 lrnperiections . whot is the prob. ,, y:i: ", ry

,ff,{r.r"a;onl,h 4 mir,ntes 7

--'-- /\-- L

\thot any one of them );lt require repcirs , what is t!-,e r,tcb.

repoirs ? , (Z)oitt olWobricoted rob.ots, er;octiy Z to require(3)ti*74-*bot is the 5h one to require repoirs ?

v-=-4 'r<:jig)Cansider the hypergeometric c;stribution vtirh : l,l=12 , o=1 ond n=3 ; vtlti:h nleens that tlie toto! numbzr t;f

u;:iis i; L2 , the n"fr5€ii;S aefeaive units is <Aia the scmple size is 3 - vvhot is the prob. af gettinE :

3-{AiA ccmLlony fXbricotes rcbLlts. tf O.1 is the prab,ttlat:(1) th; /" robot is the Jirst ane io reguirerepairs i X=j$ .^-\

(l)n.o C_e!:7t,ve units ? (2|ot teost 2 dzfective units ?. Find the rtean ei.the distribsticn by tuto methoCs-?

o

=s-*ej. Jirt|.ilsprc.u: !9s1i-r1.ngi?,r.-.. . . .

cnd (3) the mean of l,=2 + j * by *to2 D( E ';6*--zI\"y'X , Y be tu.tc independent ranclom voriobles

,Find:(lJthe moment generq.ting funciion ol Z =(z)the votue "J@D (

sucha

3X+,

tho2Y oi z,

. s/A)Let X ond y

". r:f+. {z)P(o. x :hove the joint p-d-f. f(x,y)= k ,Llv=!1 , 'p)E(x fz$,fGly) a:l.ct IUlx),

x<L.,O<Y<1).-zt

0<xcy<2.til , ond

Finci:(i) f LG) ,

@ P(o <

..::/#-r,;:. - _'rl'_ -'

Ui*l.- Y{

Let tty(O): ezo' otrd.

inC:(7J Mrr,(0),where{31P(w < -B)

cr)

l{:-'.}-if.a-.:-: --.---.r..-:i,::r; ir;.;:rj?i:.It*2;S.:.;.;qts:r?-$-?-t*_7Jg.T.45|.A?.vf-ge.:W.;5t;r:;:,,fl-q,.-";i..r.._--..._--:.._?.-:r:--;1</.?-,1,Tr.r4-;

' l-q.-\2-"-\2

y''lvrt,'cl=--*f ) Bflts1./s=,.'.? 2- -\.

e;fL"n) Mr-ic h- of ':Brq 5Je/s t-s

;

6t"n7

e

' *c*i,

o6

I

P (M - 6l = -K/41

o'.1 'P(A/ - g)

=.o-Y

- r(N //E)

'o.-3 = o'\

@ N"rllt- ^'L'I- -.-

-" 'i

PbtiB) =

0 'g iu.*+-./-

-Dl

p(tlo)-=

o-V. / t;l -

t=

O 64Y C,'/;us @ Nu r/ r'c /, "X!/- €) q^e,.fl,.. c,./q

@ rluniCL ji/e** ryf ,'/ A'.s 4-.^l-r:Z-ssets ?tu\-/- +^-,- 4

P(^'t)-.-T ? ('Li -- " 'Lt

P( P\ oB).?) So\r^ c. \

' ,lLt_qB) = PttT) r/( 6)crE : o-a t o-y P(H/18)

P(r4nS) t o-1 +o-t-{ ---B. o'l

* rWfrsl = t- ((rtUB)

<>-8 ='o'Z

Phrl -?(HnB)'9

/-P$)a.

o-n- o-J

l'L -'t)

t

--

aJ \.--,,'o\

tt

S o.*t\e +

gr/.u Po"d

//u*,y +

? d) @:r(rzq^ <s6)

rlt ?

P.

F,^J yL- i*.,-* a f---J tu- s\.^'','d*il J e''/io{'o^-

P:l .'^qq. 1*''-"'/es lo--,-f, / ".uLa^-L

o P (|r-/l,, t*-Lt"--t

:.

) d'

(oo 2

, A-7

o-o1 -: 7'o 7d

-Lt

(oo Porfs 5/L

n), f,1c. h',-z

Pfo- b

,,n 7 r*4DU--

d' Bir**-&

B'**J F0 x) { 7'-'ttP =-o17. r "-lf,

Lcr.u e'v'-b"^*d

36t( llNlz'-/*(:1*-

N

-l \-7

6-oo*o-ogT I o P(tz-<r ------------, -.--I (^ , ,,

>3.,) <rl'7' I

t'\ f,-r c'- = 3o

,, li', y = j:_/ -i'i'{-'a?t-

I @;-1o2,

? L/: Ko') t

G) f U!- L'-1*-a

--'---

LJ

= ./ 't Ka'36 -,/1

>Y.-y'1 *,*

< /-x)>/-)-- t ./ (2_t,TO'J: .--.\

V =< l-.s )

'-J

I

.--:f)t

nve,at ) trnr--'

\o -.nn i.iqt-pQ.issl.^-:--

t2- TerLca

+

Y-

d\^J is'

t.^ 1-t q". irr.^rF-e- S

N"s t /-o'\------.,-------

t^Per

)

_s-Ld:5 T-_) 'lo,w-,i*- P ( K 52) =?

:.r.Ln- \ r \\^4\r.^--

t)

C!- etz n1t 2)-

ffiffir;::."..iq>:*:4,#7I.ax.:.--)^1;.1-.-..-!::.:Ij:ii1a|:.',!:;::l,.,!7.tw,aw@@ifuP.8*

?r"\" -+ s p"tt-; j--\--\tu

(^ -r I n=o, l,tr--

5+o-\ -E

: "'j-':11 "'

a.t3g3 + o-27"( r6-Z7o(o'6n6s

f (r.) t ?r,)

(r)' ."-,

\r r--- r

GiS5,t2(") r

(2)" -2;l- et

ol- =l

4t .t2l -t

-f=

t) '=

-:

o<-T.

\FJ;-\qJ ?=

Qr"V

oJ

o-|a

"*+

-F

J f t

0

{ c4 tlivv

1 u t- i;l-v'.rrts

.

.. /(>

?=-\_\,^t.-...--

. s-l

K=L

G.-*. Vi c

- Iv.1' fet"\-s is

1 t ?. t rra c"J yc, l,.v s ,

-llrr'oL,f ry.b' o*

r :t <{h-t,':-'./-d---f^. 4eor^e4f,-C

',

4;--- le G; ^"*;,,.(

o@

@ t'k

Z_ t,t- l^f'l :o

' : : .' ':".'. 'j."--.1-.t-

i

? (/' -2) ,

o'l ) 7 =o'1 -, fr=1 , VZt

rt -4

x

5x

ff), f(:) -F.'.

6. lz tt

X'=-ti' ['}(x =a) =

r-l7 .PI: (" 1) ( ''l)

e X..c{/.rJ

.+-. Y "7,

+\)1 ':'

=/IoJ aFF

2 l- o rq4 v+i -€ruti,/- {<p^irS 2I

:

-,: V t"r!-L.t- J\

6251, d:l

\L- ?t-,<:E

lt

(ena;ts 7'r,ft \ ^-- +

l

|lsrafffi i: :.:,_ : - -:.+:--s.Fj?--'

@--f '^= X

..'.)r { -sl/rx+)=( , )F'.lu'----''- -t-+

' ({; ) i ?', o-ooc \ Zt5

5* -b46;'o-'^&

rc - Zal?-----a

@,uL:---

ry:! , 1--e-'l"t-? ,o- 4 .^* t

O.{-.o|ue- = t-l

S-bvc *ry:- = 3

S>L:.:

i?rr)--'_--

M: lz ,,Q =q,6_,5F = Yro d.fl.di*o- = o

P6="1(; ) r I )

(f)

--------

)

.'-(

o

@ L*/t 1

=';'-C?j>..:.'..:-:-'-.': -". ":'i."'t-'-"-' ':: r-""

,dryrn,? X ) z

- 'a''

- f ( r> z) , P(r.) a P(s )

(u (!F)

)

T8

a

Z

3

ttrl/f l * o- ol

lz9

-Z

(

:)GiI\

(I

/. -5 r ttx;ail

:- dfal r lf(t), z?(z)+ sP(sr'

G+

{vi tz t

. f(x-l)= \tJl'i -o-so'

Itz \

; o r o.Srl r 2("-2/f/)r S ( o.u/f/f)

rdu/

l? tz.

\u 1\< Yrob '1 t^a^,c"rd-Cn

\-r^v.-! Y-V N

i

*o{n)= 5jn, ,*,,

rS

fu* P*. \

o@

6P (z<

Neo ^

f,/t*t *+ f

.t'4/=&-

,a',I

I

\.\.?

;

l,rfrA(o) fo,i_1.

^tjGrp-^z*.rrc J.r2O

"tr'---iz

,a*

l/s',tS

r,@) =

5 -qe

' O7t

/- qoJ

/:frfria

\

Het'a" J

- ,- a'-', ,? ;. =F^ r tl * $'rtl - (*^r t 11' = --.'

-w€?,i 9 preli-^C C*_.lr*( 6,.,) + __, v-i,v =___,=-U,I* -, e

\=--- f ) 6t-'<)\/---,O'?-,r.n.t\ f,'q

\ -t 1 .-- J\ -L.-. \.t^tv +)

,E?ellre%,

--f; (c: )

5 -'i o +{ b

o lz-ry'<-

i

b

(s- q o)' (5-qa)'

+3zol) Zoc, _ lb

4rc'- ---

--?-\-)

+'(t)

( rcj)' e,

[t,i (t-,ta) j -Lq@])/ -.r4\ 5-qol'

:s -_w1(r q)' t

'

:I

:

-LrlL,"-* (oi=

-E(s-vo) I (s-uo) , ( s -vo)

2oo _ 16o

( t)'

--."-t-- '-'!- r',\qy-r'o*.r.Cj_ -:-\(x ) = cY :

:'.' "'..-., -i-.-i..:;

- - -i-rr,1:*.::_:_

.:,.:. -,--...: -,. .*....,...

..

oo, P,^ / gtt*/r;c z C { -2 5, ZoIIv

ou4 r

IN

? (z <x <s)

=@Lt

{

? ( z<

Pt"r- s)

:

Pcr :

P (r<

(l) ' (+))--

x<5) "

P (9)oP(,t)

: 2'-' P

= (i) - i+l

L{G^'^"\r,q--1.

\_l _.4r.{ P

ae) : ?(s)rf(q) =t{ )- (\

l- a '

-

/< ) 6zt

o -\L${ o-\o2.1 (

L:--z-

,-...-.'i'.'.-'

aa

a

t

I

,d

-

{., l/"lu"J

f"n HuTA*]5

[(/l , 17

\z --zti

f(2rsX) =? tf,!) = 2+t{'

oo. [( Y), 2+ 3 (s)

0J" k'^o-

^-/ -4/ -J'-"/

I

2 ^d N"fbJ

-

Ll 5e 8,^o^,& t'/elL*d

1 = t-\/,rf = 'l 5

h

/=,tt P6, y'

-'p -)t

@fuF) = eb'P^rol

a

'L.t 3*

=-*6

j'Ot=

sqL Gl--*@zs*b)= o' -( ! r*o=) , ?/Aeyv y6

'@1--..-:.-::..--.--_- -.-:_- --- - -

as)

C . H

a,

BTo;^i-ol

o

@

[,3 Pa P F(r,J)--x o(x(J(z

)

')

2

,/,

rI

/,

Z

d1

f/)/c

2 /(

(o(x

.PrttI

f_ o)

)P

@ P("a x<ta

. erE0/:)o.1 Y < T.)

foi^\ , l,J

aJt/2"

.,lfo^llJ9 ,=,

1.

'.J tg J) = l,

. L Ja

*r =l-;

[a-\-] f,?' O F (','11 )z os^t @ Jlft/,q) J^Jy=/.zt.o 12)J K J^J1 ={O U:

a

,

f rr, -trI It,t t - )

.y

'(r //,) . ,f ( r, Y)

f.cY)

'-/ /y) ) /( r, y)

ftrt

ftr, j) ,JJ

\ ,l'-- L'J I

J1

?(q<^< t ly=-;"?(x//)=

1,5^/yl ) =

ly,r-a-

z- f . rz 1t/t-7t- ?- Jo

-t .'n;2L*l_rz-:1"El-/

, t.),/J

t 7_/ 5/_/ (-

L

: [r J*) L l

@L(r-\)-----

r--\-z---:

) f, ([l=t3:Jl, (/)? ( r lx./ z

.o

:. !nJ")

f (q< ^

<,.lr,Vu)

ir., ='.{.') .-Jx

.J()

I

--?xllslJd

t/z-------1

9tIZ

: '/'- = lH-

(-,-4) ,{Gr)

o'. 3

Jr

o { X4z

tH '-'@ E (.xf s/")

€(x)- /nftrs.)XE (*1,r.)= if frxl V.

o

71,r=/ x.+ Jx

_ rzt\qi

I=.-i '\i-.-= o({(z_i k-xt

-1 : )\_.___

rdt

-\,-/,/

*

3r. ,'

L

/ -,-

,JJ

(=

r9t,

t1

U

I

5t

fJz

rI/t

I

L

B

!2

3q

5t

f:z

t()I

I

L

o

I

_. (- J,o. I

t'jI,l

='-

it'

)

II

o

?

='.

iI

.l)

,,1

I

L

1)

/l

)1

,1,

n

32 -J ).

6< / < 3/L)

X,)

=[;, +. 1,,

J3Jn

:..... . .. ....-'-.-.'.-;

Lr ={X +y_3.:

> C.,^.ver| +o sloJov-J- )r-

J)

I

v

)y

x

J

r)

Jx

: '- ':j" ''

&3_Ll

trBc\ p( h,<-s)

?"( *1- S) '

rc|< 5:2 1